`Advances in Fuzzy SystemsVolume 2011, Article ID 768028, 5 pageshttp://dx.doi.org/10.1155/2011/768028`
Research Article

On Fuzzy Sp-Open Sets

1Department of Mathematics, Rada'a College of Education and Science, Albida, Yemen
2Department of Mathematics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia

Received 29 December 2010; Accepted 18 February 2011

Copyright © 2011 Hakeem A. Othman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A new class of generalized fuzzy open sets in fuzzy topological space, called fuzzy sp-open sets, are introduced, and their properties are studied and the relationship between this new concept and other weaker forms of fuzzy open sets we discussed. Moreover, we introduce the fuzzy sp-continuous (resp., fuzzy sp-open) mapping and other stronger forms of sp-continuous (resp., fuzzy sp-open) mapping and establish their various characteristic properties. Finally, we study the relationships between all these mappings and other weaker forms of fuzzy continuous mapping and introduce fuzzy sp-connected. Counter examples are given to show the noncoincidence of these sets and mappings.

1. Introduction

In 1996, Dontchev and Przemski, [1] have introduced the concept of sp-open sets in general topology. In this paper, we extend the notion of sp-open sets to fuzzy topology space and study some notions based on this new concept. We further study the relation between fuzzy sp-open sets and other types of fuzzy open sets. We also introduce the concepts of fuzzy sp-continuous (resp., fuzzy sp-open) mapping, other stronger forms of fuzzy sp-continuous (resp., fuzzy sp-open) mapping, and discuss their relation with other weaker forms of fuzzy continuous mapping.

2. Preliminaries

Throughout this paper, by or simply by we mean a fuzzy topological space (fts, shorty) and means a mapping from a fuzzy topological space to a fuzzy topological space . If is a fuzzy set and is a fuzzy singleton in , then , , , denote, respectively, the neighborhood system of , the interior of , the closure of , and complement of .

Now, we recall some of the basic definitions and results in fuzzy topology.

Definition 2.1 (see [2]). A fuzzy singleton in is a fuzzy set defined by: , for and otherwise, where . The point is said to have support and value .

Definition 2.2. A fuzzy set in a X is called fuzzy open [3] (resp., Fuzzy preopen [4], Fuzzy -open [5]) set if Int cl Int (resp., Int cl , cl Int cl , cl Int ). The family of all fuzzy -open (resp., fuzzy preopen, fuzzy -open, fuzzy semiopen) sets of is denoted by (resp., , , ).

Definition 2.3 (see [4]). Let be any fuzzy set. Then,(i)is called preclosure,(ii) is called pre-Interior.The definitions of , , , and are similar.

Theorem 2.4. For any fuzzy set in a , the following statements are true:(i) and [6],(ii) [7].

Theorem 2.5 (see [3, 4]). The arbitrary union of fuzzy preopen (resp., fuzzy semiopen) sets is a fuzzy preopen (resp., fuzzy semiopen) set.

Definition 2.6. A mapping is said to be(i)fuzzy -continuous [3] if is fuzzy -open set in for each fuzzy open set in ,(ii)fuzzy semi continuous [3] if is fuzzy semiopen set in for each fuzzy open set in ,(iii)fuzzy -continuous [5] if is fuzzy -open set in for each fuzzy open set in .

3. Fuzzy SP-Open Set

Definition 3.1. A fuzzy subset of fuzzy space is called fuzzy sp-open set if . The class of all fuzzy sp-open sets in will be denoted be .
It is obvious that .

Proposition 3.2. Let be fuzzy sp -open set such that . Then, is fuzzy preopen.

Using Theorem 2.5, we can easily prove the next corollary.

Corollary 3.3. Any union of fuzzy sp -open sets is a fuzzy sp-open set.

Remark 3.4. The intersection of fuzzy sp-open sets need not be fuzzy sp-open set. This is illustrated by the following example.

Example 3.5. Let and , and be fuzzy sets of defined as Let . Clearly, is a fuzzy topology on , and by easy computation, it follows that and are fuzzy sp-open sets. But is not a fuzzy fuzzy sp-open set.

Theorem 3.6. For any fuzzy subset of a fuzzy space X, the following properties are equivalent:(i) is fuzzy sp-open,(ii).

Proof. . Let be a fuzzy sp-open, that is, . Then, we have

Definition 3.7. A fuzzy subset of fuzzy space is called fuzzy sp-closed set if . The class of all fuzzy sp-closed sets in will be denoted be .

Definition 3.8. Let any fuzzy set. Then,(i)is a fuzzy is called fuzzy sp-closure,(ii) is called fuzzy sp-Interior.By using Definitions 3.1, 3.7, and 3.8, we can prove the following theorems.

Theorem 3.9. Let and be the fuzzy sets in . Then, the following statements hold(i) is fuzzy sp-closed,(ii),(iii),(iv) is fuzzy sp-open,(v),(vi),(vii).

Theorem 3.10. For a fuzzy subset of a fuzzy space , the following statements are holding:(i),(ii),(iii),(iv).

Theorem 3.11. For any fuzzy subset of a fuzzy space X, the following statements are equivalent:(i) is fuzzy sp-closed, (ii) is fuzzy sp-open,(iii), and (iv).

Theorem 3.12. A fuzzy set in a fuzzy topology space is fuzzy sp -open if and only if for every fuzzy point , there exists a fuzzy sp -open set such that .

Proof. If is a fuzzy sp-open set, then we may take for every .
Conversely, we have and, hence, . This shows that is a fuzzy sp-open set.

From Definitions 2.2 and 3.1, the above “Implication Figure 1” illustrates the relation of different classes of fuzzy open sets.

Figure 1

Remark 3.13. The converse of these relations need not be true, in general as shown by the following examples.

Example 3.14. Let and , and be fuzzy sets of defined as Let . Clearly, is a fuzzy topological space on , and by easy computation, we can see:(i) is fuzzy sp-open set which is neither fuzzy -open set nor fuzzy preopen,(ii) is fuzzy sp-open which is not semiopen,(iii) is fuzzy sp-open set which is not fuzzy open.

Example 3.15. Let and fuzzy sets of defined as Let . Clearly, is a fuzzy topological space on , and by easy computation, it follows that is fuzzy -open set which is not fuzzy sp-open.

4. Fuzzy SP-Continuous Mapping

Definition 4.1. A mapping is said to be(i)fuzzy sp-continuous if is fuzzy sp-open set in for each fuzzy open set in ,(ii)fuzzy sp-continuous if is fuzzy sp-open set in for each fuzzy sp-open set in ,(iii)fuzzy sp⋆⋆-continuous if is fuzzy open set in for each fuzzy sp-open set in .

Theorem 4.2. For a mapping , the following statements are equivalent: (i) is fuzzy -continuous;(ii)for every fuzzy singleton in and every open set in such that , there exists a fuzzy sp -open set such that and ;(iii)for every fuzzy singleton in and every open set in such that , there exists a fuzzy sp -open set such that and ;(iv)the inverse image of each fuzzy closed set in is fuzzy sp-closed;(v) for each ;(vi) for each .

Proof. (i)(ii) Let fuzzy singleton be in and every open set in such that , there exists a fuzzy open set be in such that . Since is sp-continuous, is fuzzy sp-open and we have or .
(ii)(iii) Let fuzzy singleton be in and every fuzzy open set be in such that , there exists a fuzzy sp-open such that and . So, we have and .
(iii)(i) Let be a fuzzy open set in and let us take . This shows that . Since is a fuzzy open set, then there exists a fuzzy sp-open set such that and . This shows that . By Theorem 3.12, it follows that is fuzzy sp-open set in and hence is fuzzy sp-continuous.
(i)(iv) Let be a fuzzy closed in . This implies that is fuzzy open set. Hence, is fuzzy sp-open set in , that is, is fuzzy sp-open set in . Thus, is a fuzzy sp-closed set in .
(iv)(v) Let , then is sp-closed in , that is, .
(v)(vi) Let , put in , then so that . This gives .
(vi)(i) Let , be fuzzy open set. put and , then , that is, -closed set in , so is sp-continuous mapping.

Definition 4.3. A mapping is said to be(i)fuzzy sp-open (Fuzzy sp-closed) if is fuzzy sp-open (fuzzy sp-closed) set in for each fuzzy open (fuzzy closed) set in ,(ii)fuzzy sp-open (Fuzzy sp-closed) if is fuzzy sp-open (fuzzy sp-closed) set in for each fuzzy -open (fuzzy sp-closed) set in ,(iii)fuzzy sp⋆⋆-open (Fuzzy sp⋆⋆-closed) if is fuzzy open (fuzzy closed) set in for each fuzzy sp-open (fuzzy -closed) set in .

Remark 4.4. If is fuzzy sp-continuous mapping and is fuzzy sp-continuous mapping, then may not be a fuzzy sp-continuous mapping; this can be show by the following example.

Example 4.5. Let and and be fuzzy sets of defined as, Consider, , , and where , , and and the mapping and defined as and . It is clear that and are fuzzy sp-continuous mapping. But is not a fuzzy sp-continuous mapping. This because and is not fuzzy sp-open set, and hence is not fuzzy sp-continuous mapping.

Theorem 4.6. If is fuzzy sp -continuous mapping and is fuzzy continuous mapping, then is fuzzy sp-continuous mapping.

Proof. Let be a fuzzy set of . Then, . And because is fuzzy continuous this implies that is a fuzzy open set of and hence is a fuzzy sp-open set in . Therefore, is a fuzzy sp- continuous mapping.

From Definitions 4.1 and 4.3, we can have the above “Implication Figure 2” illustrates the relation between different classes of fuzzy sp-continuous (fuzzy semi sp-open) mappings.

Figure 2

The above “Implication Figure 3” illustrates the relation between fuzzy sp-continuous and different classes of fuzzy continuous mapping.

Figure 3

Remark 4.7. We can see the converse of these relations need not be true, in general as shown by the following examples.

Example 4.8. Let and , and fuzzy sets of defined as Consider , , and where , , and and the mapping and defined as and . It is clear that:(i) is a fuzzy sp-continuous mapping which is neither fuzzy -continuous mapping nor fuzzy semi continuous mapping,(ii) is a fuzzy -continuous mapping which is not fuzzy sp-continuous mapping,(iii) is a fuzzy sp-continuous mapping which is neither fuzzy sp-continuous mapping nor fuzzy sp⋆⋆-continuous mapping, and this is because is fuzzy sp-open in which is neither fuzzy sp-open nor fuzzy open in ,(iv)if defined as: , it is clear that is a fuzzy sp-open mapping which is neither fuzzy sp-open mapping nor fuzzy sp⋆⋆-open mapping.

Definition 4.9. A fuzzy set in an is said to be fuzzy connected if cannot be expressed as the union of two fuzzy separated sets.

Now, we can generalize the definition of fuzzy connected to define fuzzy sp-connected as follows.

Definition 4.10. A fuzzy set in a is said to be fuzzy sp-connected if and only if cannot be expressed as the union of two fuzzy sp-separated sets.

Theorem 4.11. Let be a fuzzy sp -continuous surjective mapping. If is a fuzzy sp -connected subset in then, is fuzzy connected in .

Proof. Suppose that is not connected in . Then, there exist fuzzy separated subsets and in such that .
Since is fuzzy sp-continuous surjective mapping, and are fuzzy sp-open set in and .
It is clear that and are fuzzy sp-separated in . Therefore, is not fuzzy sp-connected in , which is a contradiction!!
Hence, is fuzzy connected.

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